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A086792 Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G. 1
6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980 (list; graph; refs; listen; history; text; internal format)



The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.

Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the non-Abelian groups in the sequence up to n=500 and commented "In general, if 2^n - 1 is a Mersenne prime, then 2^(n-1)*(2^n - 1) is a perfect number and the group with presentation < x,y | x^(2^n-1) = 1, y^(2^n) = 1, y^-1 x y = x^-1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).


Table of n, a(n) for n=1..17.

Tom Leinster, Perfect numbers and groups

Tom De Medts, Attila MarĂ³ti, Perfect numbers and finite groups

Tom De Medts, Leinster groups

Sci.math, Groups analogy of perfect numbers


Cf. A000396.

Sequence in context: A249670 A009242 A032647 * A064987 A057341 A068412

Adjacent sequences:  A086789 A086790 A086791 * A086793 A086794 A086795




Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003


a(10)-a(17) added using "Leinster groups" link by Eric M. Schmidt, May 02 2014



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Last modified December 22 11:14 EST 2014. Contains 252355 sequences.